Problem: Two ducklings wander away from the nest while their mother is away. The first duckling's displacement (distance and direction) from the nest is ${\vec{d_1}} = (12,5)$. The second duckling's displacement is ${\vec{d_2}} = (13,-8)$. (Values above are given in meters.) How much farther did the second duckling wander than the first duckling?
Answer: The distance each duckling wandered is the magnitude of its displacement vector. Therefore, the expression $\| {\vec{d_2}} \| - \| {\vec{d_1}} \|$ tells us how much farther the second duckling wandered than the first. $\| {\vec{d_2}} \| - \| {\vec{d_1}} \| \approx 2.3$ meters Note: We can find the magnitude of any vector $\vec v$ using the Pythagorean theorem $\| \vec v \|^2 = x^2 + y^2$, where $x$ and $y$ are the components of $\vec v$. For the second question, let's find how far the first duckling is from the second duckling. To do that, let's think about the horizontal and vertical components of each vector. Horizontally, both ducklings wandered to the right, but the second duckling went $1$ meter farther. Therefore, the first duckling is $1$ meter to the left of the second duckling. Vertically, the first duckling wandered up $5$ meters and the second duckling wandered down $8$ meters. So the first duckling is $13$ meters up from the second duckling. Note that what we've just done was subtract the vectors component-wise: ${\vec{d_1}} - {\vec{d_2}} = {(12,5)} - {(13,-8)} = (-1, 13)$ Performing the subtraction the other way would have resulted in $(1,-13)$. This is the distance the second duckling is from the first. This means that to model the displacement between the ducklings, we need to subtract one of the vectors from the other. Applying the Pythagorean theorem, we can find the magnitudes of $(1,-13)$ or $(-1,13)$, which are the same. That is, we can find how far the ducklings are apart. $\| {\vec{d_1}} - {\vec{d_2}} \| = \| {\vec{d_2}} - {\vec{d_1}} \| \approx 13.0 \,\text{meters}$ The second duckling wandered $2.3$ meters farther than the first duckling. The ducklings are $13.0$ meters apart.